Learn Hardcore Finnish: Tämä kahvi vaikuttaa minuun nopeasti.

The verb vaikuttaa in the sense “to affect / to have an effect on someone” normally takes the illative case (the into case) for the person or thing that is affected.

minuun = into me (illative) → target of the effect
minua = partitive (some of me; often used as an object, or in feelings like pelottaa minua, I feel scared)
minulle = allative (to me / onto me; typical for indirect objects like antaa minulle = give to me)

So Tämä kahvi vaikuttaa minuun nopeasti literally means This coffee affects into me quickly, i.e. This coffee affects me quickly.
Using minua or minulle here would sound wrong to a native speaker.

What exactly is minuun grammatically, and how is it formed from minä?

minuun is the illative singular form of minä (I).

For minä, the main case forms you’ll meet are:

nominative: minä – I
genitive: minun – of me / my
partitive: minua – (partitive object / feelings etc.)
illative: minuun – into me

The illative case often has the idea of movement into something (physically or metaphorically):

taloon – into the house (from talo)
kouluun – into the school (from koulu)
minuun – into me (from minä)

With verbs like vaikuttaa (to affect), this “into” meaning becomes metaphorical: the effect goes into you → minuun.

I learned that vaikuttaa can also mean “to seem / to appear”. How do I know which meaning it has here?

The meaning of vaikuttaa is largely signaled by what case and what kind of complement follows it:

vaikuttaa minuun / asiaan / heihin (illative)
→ to affect, to influence
- Tämä kahvi vaikuttaa minuun nopeasti.
  = This coffee affects me quickly.
vaikuttaa hyvältä / mukavalta (translative, -lta/-ltä)
→ to seem, to appear (look/feel like something)
- Tämä kahvi vaikuttaa hyvältä.
  = This coffee seems good.
vaikuttaa siltä, että…
→ to seem that…
- Vaikuttaa siltä, että sataa.
  = It seems that it’s raining.

In your sentence you have vaikuttaa + illative (minuun), so the meaning is “affects / has an effect on”, not “seems”.

Why is kahvi in the form kahvi and not kahvia?

Here kahvi is a normal singular subject in the nominative case:

Tämä kahvi = this coffee (a specific coffee / cup / type)

You usually see partitive (kahvia) as:

an object: Juon kahvia. = I drink coffee.
a partitive subject in existential sentences: Pöydällä on kahvia. = There is coffee on the table.

But with a specific, pointed-out subject like Tämä kahvi (this particular coffee), the subject is nominative:

Tämä kahvi vaikuttaa minuun nopeasti.

Using kahvia here would not be natural, because you are clearly pointing to a specific coffee with tämä.

Why do we use nopeasti and not nopea in this sentence?

nopea is an adjective meaning quick, fast and it describes nouns:

nopea kahvi – a quick coffee (not very idiomatic, but grammatically fine)
nopea juna – a fast train

nopeasti is an adverb, formed by adding -sti to the adjective stem, and it describes verbs:

juoksee nopeasti – runs quickly
vaikuttaa nopeasti – affects quickly

In Tämä kahvi vaikuttaa minuun nopeasti, nopeasti modifies the verb vaikuttaa (how it affects), so the adverb form is required.

Can I change the word order, for example: Tämä kahvi vaikuttaa nopeasti minuun or Minuun tämä kahvi vaikuttaa nopeasti?

Yes, Finnish word order is flexible, but it changes the emphasis.

Tämä kahvi vaikuttaa minuun nopeasti.
- Neutral in many contexts.
- The adverb nopeasti at the end emphasizes the speed of the effect.
Minuun tämä kahvi vaikuttaa nopeasti.
- Emphasizes me as the person affected (maybe compared to others):
  As for me, this coffee affects me quickly.
Tämä kahvi vaikuttaa nopeasti minuun.
- Grammatical, but less neutral-sounding; the focus tends to fall more on minuun at the end (on whom it acts).

So yes, you can move the parts around, but the original word order is a very natural neutral choice.

Could I leave minuun out and just say Tämä kahvi vaikuttaa nopeasti?

You can, but the meaning becomes less specific.

Tämä kahvi vaikuttaa nopeasti.
= This coffee has a quick effect / works quickly.
(on people, on the body, or in general – it’s not explicit)
Tämä kahvi vaikuttaa minuun nopeasti.
= This coffee affects me quickly.

In real conversations, context sometimes makes minuun obvious, so speakers might drop it, but if you want to clearly say “on me”, it’s better to include minuun.

What is the nuance of tämä in Tämä kahvi compared with se or tuo?

Finnish has three common demonstratives:

tämä – this (near the speaker)
tuo – that (a bit further away, often visible)
se – that/it (known from context, not necessarily visible or near)

In your sentence:

Tämä kahvi vaikuttaa minuun nopeasti.
→ This coffee (the one I’m holding / drinking / right here) affects me quickly.

If you said:

Se kahvi vaikuttaa minuun nopeasti.
→ That coffee (the one we’ve been talking about) affects me quickly.
Tuo kahvi vaikuttaa minuun nopeasti.
→ That coffee over there affects me quickly. (farther away from speaker)

So tämä emphasizes physical / psychological closeness to the speaker.

Could I use a different verb, like tehota, instead of vaikuttaa? What’s the difference?

Yes, in many contexts you can also say:

Tämä kahvi tehoaa minuun nopeasti.

tehota means “to be effective, to work (as intended)”, and also takes the illative:

Lääke ei tehoa minuun. – The medicine doesn’t work on me.
Mainos tehosi hyvin. – The advertisement worked well.

vaikuttaa is broader: to affect, to have an effect, to influence (and also to seem in other constructions).

In your sentence:

vaikuttaa minuun: the coffee has an effect on me (general influence).
tehoaa minuun: the coffee works on me (does its job effectively).

Both are possible; vaikuttaa is more general, tehota is more about effectiveness.

How do you pronounce Tämä kahvi vaikuttaa minuun nopeasti?

Rough pronunciation guide (syllable breaks marked with .; stress on the first syllable of each word):

TÄ.mä – TÄ like ta in taco but shorter; ä like a in cat
KAH.vi – kah like kah in khan; vi like vee but short
VAI.kut.taa – vai like English eye; kut like koot but short; taa like tah but with a long a; the tt is a long consonant, so you “hold” the t slightly
MI.nuun – mi like me (short); nuun with a long u (like oo in food), so nuun is clearly longer than nun
NO.peas.ti – no like no; peas roughly peh-as with a short e and a; ti like tee but short

Very roughly in English-ish sounds:
TÄ-mä KAH-vi VAI-kut-taa MI-nuun NO-peas-ti

Key points:

Stress always on the first syllable of each word.
Double letters (tt, uu) mean the sound is longer.